# nLab twisted de Rham cohomology

Contents

### Context

#### Differential cohomology

differential cohomology

cohomology

# Contents

## Idea

The twisted cohomology-version of de Rham cohomology (a simple example of twisted differential cohomology):

For $X$ a smooth manifold and $H \in \Omega^3(X)$ a closed differential 3-form, the $H$ twisted de Rham complex is the $\mathbb{Z}_2$-graded vector space $\Omega^{even}(X) \oplus \Omega^{odd}(X)$ equipped with the $H$-twisted de Rham differential

$d + H \wedge(-) \;\colon\; \Omega^{even/odd}(X) \to \Omega^{odd/even}(X) \,,$

Notice that this is nilpotent, due to the odd degree of $H$, such that $H \wedge H = 0$, and the closure of $H$, $d H = 0$.

There is also the cohomology of the chain complex whose maps are just multiplication by $H$.

$H \wedge(-) \;\colon\; \Omega^{even/odd}(X) \to \Omega^{odd/even}(X) \,,$

This is also called H-cohomology (Cavalcanti 03, p. 19).

## Properties

###### Proposition

If the de Rham complex $(\Omega^\bullet(X),d_{dr})$ is formal, then for $H \in \Omega_{cl}^3(X)$ a closed differential 3-form, the $H$-twisted de Rham cohomology of $X$ coincides with its H-cohomology, for any closed 3-form $H$.

## References

### Twist in degree 3

The concept of $H_3$-twisted de Rham cohomology was introduced (in discussion of the B-field in string theory), in:

Further discussion:

### Twist in degree 1

The classical case of twisted de Rham cohomology with twists in degree 1, given by flat connections on flat line bundles and more generally on flat vector bundles), and its equivalence to sheaf cohomology with coefficients in abelian sheaves of flat sections (local systems):

Review:

• Cailan Li, Cohomology of Local Systems on $X_\Gamma$ (pdf, pdf)

• Youming Chen, Song Yang, Section 2.1 in: On the blow-up formula of twisted de Rham cohomology. Annals of Global Analysis and Geometry volume 56, pages 277–290 (2019) (arXiv:1810.09653, doi:10.1007/s10455-019-09667-8)

### Twist in any and higher degrees

In the broader context of twisted cohomology theory and the Chern-Dold character map:

Last revised on July 25, 2021 at 09:06:58. See the history of this page for a list of all contributions to it.